Adaptive type-2 fuzzy system for synchronisation and stabilisation of chaotic non-linear fractional order systems

This study proposes an adaptive interval type-2 Takagi-Sugeno-Kang (IT2 TSK) fuzzy system with a supervisory mode to control and stabilise a certain class of non-linear fractional order systems. In this study, a fractional order adaptation law is derived which adjusts the free parameters and bounds them by utilising a projection algorithm. The global Mittag-Leffler stability of the closed-loop system is proved in the sense that all the involved signals are uniformly bounded. Moreover, if the non-linear system tends to be unstable, a supervisory controller starts cooperating with the adaptive IT2 TSK fuzzy controller to guarantee the stability of the closed-loop system. In addition, a new inference mechanism for the adaptive IT2 TSK fuzzy system is introduced for which the antecedent part is chosen as a type-2 fuzzy set and the consequent parameters are represented as interval sets. According to the practical nature of the proposed inference equation, it would be applicable in online and real-time applications. Numerical simulations show the validity and effectiveness of the introduced control strategy for stabilisation and control of a general class of non-linear fractional order systems perturbed by disturbance and uncertainty.